Global Asymptotics Toward The Rarefaction Waves For A Parabolic-elliptic System Related To The Camassa-Holm Shallow Water Equation

This paper is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equationwith the initial datau(0, x) = u₀(x) → u_±, as x→ ±∞. (I)where u₁ < u₊ are two constants and f(u) is a sufficiently smooth function satisfying f"(u) > 0 for all u under consideration.Our main purpose is to study the relation between solutions to the above Cauchy problem and the rarefaction wave solutions u^R (x/t):of the corresponding nonlinear conservation laws.Let U(t, x) be the smooth approximation of the rarefaction wave profile. We show that if u₀(x) — U(0, x) ∈ H¹(R) and u_- < u₊, the above Cauchy problem (E) and (I) admits a unique global classical solution u(t, x) which tends asymptotically to the rarefaction wave u^R(x/t) in the maximum norm.